314. FDT-violation in colloidal glasses under shear
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Matthias Krueger is a PhD student of Matthias Fuchs (U. of Konstanz). His interests include soft condense matter (that’s what this post will be about) and mesoscopic physics. Dmitry.
It is a pleasure for me to follow Dmitry’s invitation and to write a post about the paper which Matthias Fuchs and I have recently written (arXiv:0903.0558). We investigated the extension of the well known fluctuation dissipation theorem (FDT) to the nonequilibrium situation of dense (glassy) Brownian particles under shear.
The FDT is a fundamental theorem of statistical physics, it connects two quantities: First, the time dependent correlation function C(t). It describes the relaxation of thermal fluctuations (in our system mostly fluctuations of the local density of particles) as function of time. For example, it measures how strongly the positions of the particles at time t are correlated to their positions at t=0. The further away the particles move from their original positions, the smaller is C(t). Second, the response 
of the system to a small external perturbation. It is connected to the “softness” of the system related to the question “How does the system change if we perturb it a little bit by an external force?”. In equilibrium, the FDT takes the following form,
.
The connection of these two quantities has a very intuitive interpretation: Knowing the “softness” of a system in response to an external force, one should be able to predict how strongly the system fluctuates at a given temperature T. Of course, softer systems fluctuate more strongly.
Let us turn to our nonequilibrium system: We consider Brownian particles (also called colloids, particles of micrometer size suspended in a solvent) at high density under shear, see the figure. At high density, the dynamics of the system without shear is very slow. Crossing the glass transition density, the particles cannot leave their immediate environment anymore, they are trapped in the “cages” built by the surrounding particles. In the figure, the magenta particle is trapped by the green ones. It is like standing in an overfull elevator, nobody can move anymore. In the glassy state, the system is nonergodic (meaning e.g. that C(t) does not decay to zero). When we now drive the system by an external shear flow, it becomes ergodic again, i.e., the particles move again and C(t) decays to zero. The dynamics is then governed completely by shear (since the particles are trapped without it) even for very small shear rates, the limit we are interested in. In this situation, the FDT is violated in a very peculiar way: At short times (corresponding to the short-time diffusion of the particles inside the cages), the movement of the particles is not yet influenced by the external shear and the equilibrium FDT holds. During the shear governed long time dynamics, the FDT is violated, but it takes the same shape as in equilibrium, only the pre-factor is different,
.
The quantity X is commonly referred to as fluctuation dissipation ratio (FDR) which is unity in equilibrium. The interesting finding here is that it is constant in time for the whole shear governed (long-time) dynamics. This was found in simulations [1] of a glassy Lennard-Jones system under shear. In the cited simulations, the FDR was also found to take the same value (
) for different observables (e.g. different kinds of external perturbations) which lead to the notion of an effective temperature 
describing the nonequilibrium state. If X is independent of observable, the system effectively behaves like an equilibrium system at a temperature given by the effective temperature.
Our theoretical study which was based on techniques from the mode coupling theory (MCT) aimed to address the following questions:
- Is the FDR X for long times constant in time?
- What value does it take?
- Does this value depend on the considered observable?
The first question is clearly to be answered by “yes” meaning that the equilibrium-shape of the FDT is indeed recovered at long times (with pre-factor X). Concerning the value, we interestingly find that X=1/2 in the simplest approximation, for all variables. This is in agreement with the describtion in terms of an effective temperature. The fact that the FDR is smaller than unity means that the system appears “stiffer” as one would expect from its fluctuations. Even more interestingly, the value X=1/2 shows up at many places in the literature concerning the FDT violation of critical spin models. The connection of these models to our system is still unclear, as well as the physical meaning of this special value. Looking more closely, there are corrections to this universal value in our calculations, and these corrections depend on the considered observable. They therefore contradict the notion of an effective temperature.
Further reading
[1] L. Berthier and J.-L. Barrat, J. Chem. Phys. 116, 6228 (2002).
[2] J.-P. Hansen and I. R. McDonald. Theory of Simple Liquids ? 3nd ed., good book on soft condensed matter.
[3] Soft and fragile Matter. Edited by M. E. Cates and M. R. Evans (Scottish Universities Summer School in Physics & Institute of Physics Publishing, Bristol & Philadelphia, 2000). Another good textbook.
[4] R. G. Larson. The Structure and Rheology of Complex Fluids (Oxford University Press, New York, 1999). A very good book useful for anyone interested in complex fluids. Very clear and concise.
[5] K. G. Dhont. An Introduction to Dynamics of Colloids (Elsevier science, Amsterdam, 1996). Excellent book on colloids.
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